PG26839.B00213
[PDF page 212] 196 THE MATHEMATICAL TRIPOS. [CH. VII a degree. In 1858 the fiction that the poll-examinations were part of the Senate-House examination was abandoned, and subsequently they have been treated as providing an independent method of obtaining the degree: thus now the mathematical tripos is the sole representative of the old Senate-House examination. Since 1858 numerous other ways of obtaining the degree have been established, and it is now possible to get it by shewing proficiency in very special, or even technical subjects. Further changes in the mathematical tripos were introduced in 1833*. The duration of the examination, before the issue of the brack- ets, was extended to five days, and the number of hours of examination on each day was fixed at five and a-half. Seven and a-half hours were assigned to problems. The examination on the first day was confined to subjects that did not require the differential calculus, and only the simplest applications of the calculus were permitted on the second and third days. During the first four days of the examination the same papers were set to all the candidates alike, but on the fifth day the ex- amination was conducted according to classes. No reference was made to vivâ vocequestions, and the preliminary classification of the brackets only survived in a permission to re-examine candidates if it were found necessary. This permissive rule remained in force till 1848, but I believe that in fact it was never used. In December, 1834, a few unimportant details were amended. Mr Earnshaw, the senior moderator in 1836, informed me that he believed that the tripos of that year was the earliest one in which all the papers were marked, and that in previous years the examiners had partly relied on their impression of the answers given. New regulations came into force† in 1839. The examination now lastedforsixdays, andcontinuedasbeforeforfivehoursanda-halfeach day. Eight and a-half hours were assigned to problems. Throughout the whole examination the same papers were set to all candidates, and no reference was made to any preliminary classes. It was no doubt in accordance with the spirit of these changes that the acts in the schools should be abolished, but they were discontinued by the moderators of 1839 without the authority of the Senate. The examination was for the future confined‡ to mathematics. * See the Grace of April 6, 1832. † See Grace of May 30, 1838. ‡ Under a badly-worded grace passed on May 11, 1842, on the recommendation of

PG26839.B00214
[PDF page 213] CH. VII] THE MATHEMATICAL TRIPOS. 197 In the same year in which the new scheme came into force a pro- posal to again reopen the subject was rejected (March 6). The difficulty of bringing professorial lectures into relation with the needs of students has more than once been before the Univer- sity. The desirability of it was emphasized by a Syndicate in February, 1843, which recommended conferences at stated intervals between the mathematical professors and examiners. This report foreshadowed the creation of a Mathematical Board, but it was rejected by the Senate on March 31. A few years later the scheme of the examination was again recon- structed by regulations* which came into effect in 1848. The dura- tion of the examination was extended to eight days. The examination lasted in all forty-four and a-half hours, twelve of which were devoted to problems. The first three days were assigned to specified elementary subjects; in the papers set on these days riders were to be set as well as bookwork, but the methods of analytical geometry and the calculus were excluded. After the first three days there was a short interval, at the end of which the examiners issued a list of those who had so acquit- ted themselves as to deserve mathematical honours. Only those whose names were contained in this list were admitted to the last five days of the examination, which was devoted to the higher parts of mathe- matics. After the conclusion of the examination the examiners, taking into account the whole eight days, brought out the list arranged in or- der of merit. No provision was made for any rearrangement of this list corresponding to the examination of the brackets. The arrangements of 1848 remained in force till 1873. In the same year as these regulations came into force, a Board of Mathematical Studies (consisting of the mathematical professors, and the moderators and examiners for the current year and the two pre- ceding years) was constituted†by the Senate. From that time forward their minutes supply a permanent record of the changes gradually in- troduced into the tripos. I do not allude to subsequent changes which only concern unimportant details of the examination. a syndicate on theological studies, candidates for mathematical honours were, after 1846, required to attend the poll examination on Paley’s Moral Philosophy, the New Testament and Ecclesiastical History. This had not been the intention of the Senate, and on March 14, 1855, a grace was passed making this clear. * See Grace of May 13, 1846, confirming a report of March 23, 1846. † See Grace of October 31, 1848.

PG26839.B00215
[PDF page 214] 198 THE MATHEMATICAL TRIPOS. [CH. VII In May, 1849, the Board issued a report in which, after giving a review of the past and existing state of the mathematical studies in the University, they recommended that the mathematical theories of electricity, magnetism, and heat should not be admitted as subjects of examination. In the following year they issued a second report, in which they recommended the omission of elliptical integrals, Laplace’s coefficients, capillary attraction, and the figure of the earth considered as heterogeneous, as well as a definite limitation of the questions in lunar and planetary theory. In making these recommendations the Board were only giving expression to what had become the practice in the examination. I may, in passing, mention a curious attempt which was made in 1853 and1 1854 to assist candidates in judging of the relative difficulty of the questions asked. This was effected by giving to the candidates, at the same time as the examination paper, a slip of paper on which the marks assigned for the bookwork and rider for each question were printed. I mention the fact merely because these things are rapidly for- gotten and not because it is of any intrinsic value. I possess a complete set of slips which came to me from Dr Todhunter. In 1856 there was an amusing difference of opinion between the Vice-Chancellor and the moderators. The Vice-Chancellor issued a notice to say that for the convenience of the University he had directed the tripos lists to be published at 8.0 a.m. as well as at 9.0 a.m., but when the University arrived at 8.0 the moderators said that they should not read the list until 9.0. Considerablechangesintheschemeofexaminationwereintroduced in 1873. On December 5, 1865, the Board had recommended the ad- dition of Laplace’s coefficients and the figure of the earth considered as heterogeneous as subjects of the examination; the report does not seem to have been brought before the Senate, but attention was called to the fact that certain departments of mathematics and mathematical physics found no place in the tripos schedules, and were neglected by most students. Accordingly a syndicate was appointed on June 6, 1867, to consider the matter, and a scheme drawn up by them was approved in 1868* and came into effect in 1873. The new scheme of examination * See Grace of June 2, 1868. It was carried by a majority of only five in a house of 75. 1. ‘1853 and’ inserted as per errata sheet

PG26839.B00216
[PDF page 215] CH. VII] THE MATHEMATICAL TRIPOS. 199 was framed on the same lines as that of 1848. The subjects in the first three days were left unchanged, but an extra day was added, devoted to the elements of mathematical physics. The essence of the modifi- cation was the greatly extended range of subjects introduced into the schedule of subjects for the last five days, and their arrangement in divisions, the marks awarded to the five divisions being approximately those awarded to the three days in proportion to2, 1, 1, 1, 2/3 to 1 respectively. Under the new regulations the number of examiners was increased from four to five. The assignment of marks to groups of subjects was made under the impression that the best candidates would concentrate their abilities on a selection of subjects from the various divisions. But it was found that, unless the questions were made extremely difficult, more marks could be obtained by reading superficially all the subjects in the five divisions than by attaining real proficiency in a few of the higher ones: while the wide range of subjects rendered it practically impossible to thoroughly cover all the ground in the time allowed. The failure was so pronounced that in 1877 another syndicate was appointed to consider the mathematical studies and examinations of the University. They presented an elaborate scheme, but on May 13, 1878, some of the most important parts of it were rejected and their subsequent proposals, acceptedonNovember21, 1878(by62to49), representedacompromise which pleased few members of the Senate*. Under the new scheme which came into force in 1882 the tripos was divided into two portions: the first portion was taken at the end of the third year of residence, the range of subjects being practically the same as in the regulations of 1848, and the result brought out in the customary order of merit. The second portion was held in the following January, and was open only to those who had been wranglers in the preceding June. This portion was confined to higher mathematics and appealed chiefly to specialists. The result was brought out in three classes, each arranged in alphabetical order. The moderators and ex- aminers conducted the whole examination without any extraneous aid. In the next year or two further amendments were made†, moving * See Graces of May 17, 1877; May 29, 1878; and November 21, 1878: and the Cambridge University Reporter, April 2, May 14, June 4, October 29, Novem- ber 12, and November 26, 1878. † See the Graces of December 13, 1883; June 12, 1884; February 10, 1885; Octo- ber 29, 1885; and June 1, 1886.

PG26839.B00217
[PDF page 216] 200 THE MATHEMATICAL TRIPOS. [CH. VII the second part to the June of the fourth year, throwing it open to all men who had graduated in the tripos of the previous June, and transferring the conduct of the examination in Part 2 to four examiners nominated by the Board: this put it largely under the control of the professors. The range of subjects of Part 2 was also greatly extended, and candidates were encouraged to select only a few of them. It was further arranged that Part 1 might be taken at the end of a man’s second year of residence, though in that case it would not qualify for a degree. A student who availed himself of this leave could take Part 2 at the end either of his third or of his fourth year as he pleased. The tripos is still (1905) carried on under the scheme of 1886. The general effect of these changes was to destroy the homogeneity of the tripos. Objections to the new scheme were soon raised. Espe- cially, it was said—whether rightly or wrongly—that Part 1 contained too many technical subjects to serve as a general educational training for any save mathematicians; that the distinction of a high place in the historic list produced on its results tended to prevent the best men taking it in their second year, though by this time they had read suf- ficiently to be able to do so; and that Part 2 was so constructed as to appeal only to professional mathematicians, and that thus the higher branches of mathematics were neglected by all save a few specialists. Whatever value be attached to these opinions, the number of stu- dents studying mathematics fell rapidly under the scheme of 1886. In 1899theBoardproposed * furtherchanges. Theseseemedtosomemem- bers of the Senate to be likely to still further decrease the number of men who took up the subject as one of general education. At any rate the two main proposals were rejected (February 15, 1900) by votes of 151 to 130 and 161 to 129. The curious origin of the term tripos has been repeatedly told, and an account of it may fitly close this chapter. Formerly there were three principal occasions on which questionists were admitted to the title or degree of bachelor. The first of these was the comitia priora, held on Ash-Wednesday, for the best men in the year. The next was the comitia posteriora, which was held a few weeks later, and at which any student who had distinguished himself in the quadragesimal exercises subsequenttoAsh-Wednesdayhadhisseniorityreservedtohim. Lastly, there was the comitia minora, for students who had in no special way * See Reports dated November 7, 1899, and January 20, 1900.

PG26839.B00218
[PDF page 217] CH. VII] THE MATHEMATICAL TRIPOS. 201 distinguished themselves. In the fifteenth century an important part in the ceremony on each of these occasions was taken by a certain “ould bachilour,” who sat upon a three-legged stool or tripos before the proctors and tested the abilities of the would-be graduates by arguing some question with the “eldest son,” who was selected from them as their representative. To assist the latter in what was often an unequal contest his “father,” that is, the officer of his college who was to present him for his degree, was allowed to come to his assistance. Originally the ceremony was a serious one, and had a certain reli- gious character. It took place in Great St Mary’s Church, and marked the admission of the student to a position with new responsibilities, while the season of Lent was chosen with a view to bring this into prominence. The Puritan party objected to the observance of such ec- clesiastical ceremonies, and in the course of the sixteenth century they introduced much license and buffoonery into the proceedings. The part played by the questionist became purely formal. A serious debate still sometimes took place between the father of the senior questionist and a regent master who represented the University; but the discussion was prefaced by a speech by the bachelor, who came to be called Mr Tripos just as we speak of a judge as the bench, or of a rower as an oar. Ulti- mately public opinion permitted Mr Tripos to say pretty much what he pleased, so long as it was not dull and was scandalous. The speeches he delivered or the verses he recited were generally preserved by the Reg- istrary, and were known as the tripos verses: originally they referred to the subjects of the disputations then propounded. The earliest copies now extant are those for 1575. The University officials, to whom the personal criticisms in which the tripos indulged were by no means pleasing, repeatedly exhorted him to remember “while exercising his privilege of humour, to be modest withal.” In 1740, says Mr Mullinger*, “the authorities after condemn- ing the excessive license of the tripos announced that the comitia at Lent would in future be conducted in the Senate-House; and all mem- bers of the University, of whatever order or degree, were forbidden to assail or mock the disputants with scurrilous jokes or unseemly witti- cisms. About the year 1747–8, the moderators initiated the practice of printing the honour lists on the back of the sheets containing the * J.B. Mullinger, The University of Cambridge, Cambridge, vol. i, 1873, pp. 175, 176.

PG26839.B00219
[PDF page 218] 202 THE MATHEMATICAL TRIPOS. [CH. VII tripos verses, and after the year 1755 this became the invariable prac- tice. By virtue of this purely arbitrary connection these lists themselves became known as the tripos; and eventually the examination itself, of which they represented the results, also became known by the same designation.” The tripos ceased to deliver his speech about 1750, but the issue of tripos verses continued for nearly 150 years longer. During the latter part of this time they consisted of four sets of verses, usually in Latin, butoccasionallyinGreek, inwhichcurrenttopicsintheUniversitywere treated lightly or seriously as the writer thought fit. They were written for the proctors and moderators by undergraduates or commencing bachelors, who were supposed each to receive a pair of white kid gloves in recognition of their labours. Thus gradually the word tripos changed its meaning “from a thing of wood to a man, from a man to a speech, from a speech to sets of verses, from verses to a sheet of coarse foolscap paper, from a paper to a list of names, and from a list of names to a system of examination*.” In 1895 the proctors and moderators, without consulting the Sen- ate, sent in no verses, and thus, in spite of widespread regret, an in- teresting custom of many centuries standing was destroyed. No doubt it may be argued that the custom had never been embodied in statute or ordinance, and thus was not obligatory. Also it may be said that its continuance was not of material benefit to anybody. I do not think that such arguments are conclusive, and personally I regret the disap- pearance of historic ties unless it can be shown that they cause incon- venience, which of course in this case could not be asserted. * Wordsworth, Scholae Academicae, Cambridge, 1877, p. 21.

PG26839.B00220
[PDF page 219] CHAPTER VIII. THREE GEOMETRICAL PROBLEMS. Among the more interesting geometrical problems of antiquity are three questions which attracted the special attention of the early Greek mathematicians. Our knowledge of geometry is derived from Greek sources, and thus these questions have attained a classical position in the history of the subject. The three questions to which I refer are (i) the duplication of a cube, that is, the determination of the side of a cube whose volume is double that of a given cube; (ii) the trisection of an angle; and (iii) the squaring of a circle, that is, the determination of a square whose area is equal to that of a given circle—each problem to be solved by a geometrical construction involving the use of straight lines and circles only, that is, by Euclidean geometry. With the restriction last mentioned all three problems are insol- uble†. To duplicate a cube the length of whose side isa, we have to find a line of lengthx, such that x3 = 2 a3. Again, to trisect a given angle, we may proceed to find the sine of the angle, saya, then, if x is the sine of an angle equal to one-third of the given angle, we have 4x3 = 3 x − a. Thus the first and second problems, when con- sidered analytically, require the solution of a cubic equation; and since a construction by means of circles (whose equations are of the form x2 + y2 + ax + by + c = 0) and straight lines (whose equations are of the form αx + βy + γ = 0) cannot be equivalent to the solution of a cubic equation, it is inferred that the problems are insoluble if in our † F. Klein, Vorträge über ausgewählte Fragen der Elementargeometrie, Leipzig, 1895. 203

PG26839.B00221
[PDF page 220] 204 THREE GEOMETRICAL PROBLEMS. [CH. VIII constructions we are restricted to the use of circles and right lines. If the use of the conic sections is permitted, both of these questions can be solved in many ways. The third problem is different in character, but under the same restrictions it also is insoluble. I propose to give some of the constructions which have been pro- posed for solving the first two of these problems. To save space, I shall not draw the necessary diagrams, and in most cases I shall not add the proofs: the latter present but little difficulty. I shall conclude with some historicalnotesonapproximatesolutionsofthequadratureofthecircle. The Duplication of the Cube*. The problem of the duplication of the cube was known in ancient timesastheDelianproblem, inconsequenceofalegendthattheDelians had consulted Plato on the subject. In one form of the story, which is related by Philoponus†, it is asserted that the Athenians in 430b.c., when suffering from the plague of eruptive typhoid fever, consulted the oracle at Delos as to how they could stop it. Apollo replied that they must double the size of his altar which was in the form of a cube. To the unlearned suppliants nothing seemed more easy, and a new altar was constructed either having each of its edges double that of the old one (from which it followed that the volume was increased eight-fold) or by placing a similar cubic altar next to the old one. Whereupon, according to the legend, the indignant god made the pestilence worse than before, and informed a fresh deputation that it was useless to trifle with him, as his new altar must be a cube and have a volume exactly double that of his old one. Suspecting a mystery the Athenians applied to Plato, who referred them to the geometricians. The insertion of Plato’s name is an obvious anachronism. Eratosthenes‡ relates a somewhat similar story, but with Minos as the propounder of the problem. * See Historia Problematis de Cubi DuplicationebyN.T.Reimer, Göttingen, 1798; and Historia Problematis Cubi Duplicandiby C.H. Biering, Copenhagen, 1844: also Das Delische Problem, by A. Sturm, Linz, 1895–7. Some notes on the subject are given in myHistory of Mathematics. † Philoponus ad Aristotelis Analytica Posteriora, bk. i, chap. vii. ‡ Archimedis Opera cum Eutocii Commentariis, ed. Torelli, Oxford, 1792, p. 144; ed. Heiberg, Leipzig, 1880–1, vol.iii, pp. 104–107.

PG26839.B00222
[PDF page 221] CH. VIII] THE DUPLICATION OF THE CUBE. 205 In an Arab work, the Greek legend was distorted into the following extraordinarily impossible piece of history, which I cite as a curiosity of its kind. “Now in the days of Plato,” says the writer, “a plague broke out among the children of Israel. Then came a voice from heaven to one of their prophets, saying, ‘Let the size of the cubic altar be doubled, and the plague will cease’; so the people made another altar like unto the former, and laid the same by its side. Nevertheless the pestilence continued to increase. And again the voice spake unto the prophet, saying, ‘They have made a second altar like unto the former, and laid it by its side, but that does not produce the duplication of the cube.’ Then applied they to Plato, the Grecian sage, who spake to them, saying, ‘Ye have been neglectful of the science of geometry, and therefore hath God chastised you, since geometry is the most sublime of all the sciences.’ Now, the duplication of a cube depends on a rare problem in geometry, namely...”. And then follows the solution of Apollonius, which is given later. If a is the length of the side of the given cube andx that of the required cube, we havex3 = 2a3, that is,x : a = 3√ 2 : 1. It is probable that the Greeks were aware that the latter ratio is incommensurable, in other words, that no two integers can be found whose ratio is the same as that of 3√ 2 : 1, but it did not therefore follow that they could not find the ratio by geometry: in fact, the side and diagonal of a square are instances of lines whose numerical measures are incommensurable. I proceed now to give some of the geometrical constructions which have been proposed for the duplication of the cube*. With one ex- ception, I confine myself to those which can be effected by the aid of the conic sections. Hippocrates†(circ. 420b.c.) was perhaps the earliest mathemati- cian who made any progress towards solving the problem. He did not give a geometrical construction, but he reduced the question to that of finding two means between one straight line (a), and another twice as long (2a). If these means arex and y, we havea : x = x : y = y : 2a, from which it follows thatx3 = 2a3. It is in this form that the problem * On the application to this problem of the traditional Greek methods of analysis by Hero and Philo (leading to the solution by the use of Apollonius’s circle), by Nicomedes (leading to the solution by the use of the conchoid), and by Pappus (leading to the solution by the use of the cissoid), seeGeometrical Analysisby J. Leslie, Edinburgh, second edition, 1811, pp. 247–250, 453. † Proclus, ed. Friedlein, pp. 212, 213.

PG26839.B00223
[PDF page 222] 206 THREE GEOMETRICAL PROBLEMS. [CH. VIII is always presented now. Formerly any process of solution by finding these means was called a mesolabum. One of the first solutions of the problem was that given by Archy- tas* in or about the year 400b.c. His construction is equivalent to the following. On the diameterOA of the base of a right circular cylinder describe a semicircle whose plane is perpendicular to the base of the cylinder. Let the plane containing this semicircle rotate round the gen- erator throughO, then the surface traced out by the semicircle will cut the cylinder in a tortuous curve. This curve will itself be cut by a right cone, whose axis isOA and semi-vertical angle is (say)60◦, in a point P, such that the projection ofOP on the base of the cylinder will be to the radius of the cylinder in the ratio of the side of the required cube to that of the given cube. Of course the proof given by Archytas is geo- metrical; and it is interesting to note that in it he shows himself familiar with the results of the propositions Euc.iii, 18,iii, 35, andxi, 19. To show analytically that the construction is correct, takeOA as the axis of x, and the generator of the cylinder drawn throughO as axis of z, then with the usual notation, in polar coordinates, ifa is the radius of the cylinder, we have for the equation of the surface described by the semicircle r = 2a sin θ; for that of the cylinderr sin θ = 2a cos φ; and for that of the conesin θ cos φ = 1 2. These three surfaces cut in a point such that sin3 θ = 1 2, and therefore(r sin θ)3 = 2a3. Hence the volume of the cube whose side isr sin θ is twice that of the cube whose side isa. The construction attributed to Plato†(circ. 360b.c.) depends on the theorem that, if CAB and DAB are two right-angled triangles, having one side,AB, common, their other sides,AD and BC, parallel, and their hypothenuses, AC and BD, at right angles, then if these hypothenuses cut in P, we have P C : P B = P B : P A = P A : P D. Hence, if such a figure can be constructed having P D = 2 P C, the problem will be solved. It is easy to make an instrument by which the figure can be drawn. The next writer whose name is connected with the problem is Menaechmus‡, who in or about 340b.c. gave two solutions of it. In the first of these he pointed out that two parabolas having a common vertex, axes at right angles, and such that the latus rectum * Archimedis Opera, ed. Torelli, p. 143; ed. Heiberg, vol.iii, pp. 98–103. † Ibid., ed. Torelli, p. 135; ed. Heiberg, vol.iii, pp. 66–71. ‡ Ibid., ed. Torelli, pp. 141–143; ed. Heiberg, vol.iii, pp. 92–99.
